Theorem 1 (Chain rule) Let , , where and are open in , such that are differentiable on their respective domains. Then is also differentiable on , with for all .
Proof: We first assume that there exists a neighborhood of for which . This happens in the case of by inverse function theorem. In that case, by the definition of derivative and its properties, we have
In the case of , we have that for all ,
From this, we easily verifies that , which means that is differentiable at and in the case of , must hold as well.
Lemma 1 Let , be differentiable with and . Then,
Instead of , one can also use any closed interval of .
Proof: Follows directly from Fundamental Theorem of Calculus. See Theorem 2 (Newton-Leibniz axiom) of .
Lemma 1 is a statement of invariance of integral along parameterized smooth paths with the same endpoints.
Theorem 2 (Change of variables or u-substitution in integration) Let be any differentiable function of on , which is continuous on , and be Riemann integrable on intervals in its domain. Then,
Proof: Let be an antiderivative of . By the Fundamental Theorem of Calculus, it suffices to show that the left hand side of is equal to , which can be done by applying Lemma 1 accordingly.
Theorem 3 (Integration by parts) Let be differentiable functions on and continuous on . Then,
Proof: We have
Rearranging the above completes the proof.